3.4.9 \(\int (g+h x)^2 (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3 \, dx\) [309]

Optimal. Leaf size=875 \[ -\frac {B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}+\frac {B^2 (b c-a d)^2 h^2 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b^3 d^2}-\frac {2 B^2 (b c-a d)^2 h (3 b d g-2 b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b^3 d^3}-\frac {B (b c-a d) h (3 b d g-2 b c h-a d h) n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b^3 d^2}-\frac {B (b c-a d) h^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b d^3}+\frac {B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b^3 d^3}-\frac {(b g-a h)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 b^3 h}+\frac {(g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 h}-\frac {B^2 (b c-a d)^3 h^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-2 b c h-a d h) n^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}+\frac {B^3 (b c-a d)^3 h^2 n^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 d^3}-\frac {2 B^3 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^3 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3} \]

[Out]

-B^3*(-a*d+b*c)^3*h^2*n^3*ln(d*x+c)/b^3/d^3+B^2*(-a*d+b*c)^2*h^2*n^2*(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))
/b^3/d^2-2*B^2*(-a*d+b*c)^2*h*(-a*d*h-2*b*c*h+3*b*d*g)*n^2*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+
c)^n)))/b^3/d^3-B*(-a*d+b*c)*h*(-a*d*h-2*b*c*h+3*b*d*g)*n*(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b^3/d^2-
1/2*B*(-a*d+b*c)*h^2*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b/d^3+B*(-a*d+b*c)*(a^2*d^2*h^2-a*b*d*h*(
-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/
b^3/d^3-1/3*(-a*h+b*g)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/b^3/h+1/3*(h*x+g)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^
n)))^3/h-B^2*(-a*d+b*c)^3*h^2*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*ln(1-b*(d*x+c)/d/(b*x+a))/b^3/d^3-2*B^3*(-
a*d+b*c)^2*h*(-a*d*h-2*b*c*h+3*b*d*g)*n^3*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^3+2*B^2*(-a*d+b*c)*(a^2*d^2*h^2
-a*b*d*h*(-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*polylog(2,d*(b*
x+a)/b/(d*x+c))/b^3/d^3+B^3*(-a*d+b*c)^3*h^2*n^3*polylog(2,b*(d*x+c)/d/(b*x+a))/b^3/d^3-2*B^3*(-a*d+b*c)*(a^2*
d^2*h^2-a*b*d*h*(-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n^3*polylog(3,d*(b*x+a)/b/(d*x+c))/b^3/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.12, antiderivative size = 875, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 16, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {2573, 2553, 2398, 2404, 2339, 30, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \begin {gather*} -\frac {B^3 h^2 n^3 \log (c+d x) (b c-a d)^3}{b^3 d^3}-\frac {B^2 h^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) (b c-a d)^3}{b^3 d^3}+\frac {B^3 h^2 n^3 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) (b c-a d)^3}{b^3 d^3}+\frac {B^2 h^2 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (b c-a d)^2}{b^3 d^2}-\frac {2 B^2 h (3 b d g-2 b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (b c-a d)^2}{b^3 d^3}-\frac {2 B^3 h (3 b d g-2 b c h-a d h) n^3 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^2}{b^3 d^3}-\frac {B h^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)}{2 b d^3}-\frac {B h (3 b d g-2 b c h-a d h) n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)}{b^3 d^2}+\frac {B \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)}{b^3 d^3}+\frac {2 B^2 \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{b^3 d^3}-\frac {2 B^3 \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n^3 \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{b^3 d^3}-\frac {(b g-a h)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 b^3 h}+\frac {(g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 h} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3,x]

[Out]

-((B^3*(b*c - a*d)^3*h^2*n^3*Log[c + d*x])/(b^3*d^3)) + (B^2*(b*c - a*d)^2*h^2*n^2*(a + b*x)*(A + B*Log[(e*(a
+ b*x)^n)/(c + d*x)^n]))/(b^3*d^2) - (2*B^2*(b*c - a*d)^2*h*(3*b*d*g - 2*b*c*h - a*d*h)*n^2*Log[(b*c - a*d)/(b
*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b^3*d^3) - (B*(b*c - a*d)*h*(3*b*d*g - 2*b*c*h - a*d*h
)*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(b^3*d^2) - (B*(b*c - a*d)*h^2*n*(c + d*x)^2*(A + B*
Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b*d^3) + (B*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*
d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(
b^3*d^3) - ((b*g - a*h)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(3*b^3*h) + ((g + h*x)^3*(A + B*Log[(e*(
a + b*x)^n)/(c + d*x)^n])^3)/(3*h) - (B^2*(b*c - a*d)^3*h^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1
- (b*(c + d*x))/(d*(a + b*x))])/(b^3*d^3) - (2*B^3*(b*c - a*d)^2*h*(3*b*d*g - 2*b*c*h - a*d*h)*n^3*PolyLog[2,
(d*(a + b*x))/(b*(c + d*x))])/(b^3*d^3) + (2*B^2*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^
2*g^2 - 3*c*d*g*h + c^2*h^2))*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*
x))])/(b^3*d^3) + (B^3*(b*c - a*d)^3*h^2*n^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*d^3) - (2*B^3*(b*c
- a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n^3*PolyLog[3, (d*(a + b*
x))/(b*(c + d*x))])/(b^3*d^3)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2355

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[x*((a + b*Log[c*x^n])
^p/(d*(d + e*x))), x] - Dist[b*n*(p/d), Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2398

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol]
:> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Dist[b*n*(p/((q
+ 1)*(e*f - d*g))), Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{
a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2553

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[b*c - a*d, Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m +
2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && Inte
gerQ[m] && IGtQ[p, 0]

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx &=\int \left (A^3 (g+h x)^2+3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac {A^3 (g+h x)^3}{3 h}+\left (3 A^2 B\right ) \int (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+\left (3 A B^2\right ) \int (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^3 \int (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac {A^3 (g+h x)^3}{3 h}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {\left (A^2 B (b c-a d) n\right ) \int \frac {(g+h x)^3}{(a+b x) (c+d x)} \, dx}{h}-\frac {\left (2 A B^2 (b c-a d) n\right ) \int \frac {(g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}-\frac {\left (B^3 (b c-a d) n\right ) \int \frac {(g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=\frac {A^3 (g+h x)^3}{3 h}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {\left (A^2 B (b c-a d) n\right ) \int \left (\frac {h^2 (3 b d g-b c h-a d h)}{b^2 d^2}+\frac {h^3 x}{b d}+\frac {(b g-a h)^3}{b^2 (b c-a d) (a+b x)}+\frac {(d g-c h)^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{h}-\frac {\left (2 A B^2 (b c-a d) n\right ) \int \left (\frac {h^2 (3 b d g-b c h-a d h) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d^2}+\frac {h^3 x \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {(b g-a h)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 (b c-a d) (a+b x)}+\frac {(d g-c h)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{h}-\frac {\left (B^3 (b c-a d) n\right ) \int \left (\frac {h^2 (3 b d g-b c h-a d h) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d^2}+\frac {h^3 x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {(b g-a h)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 (b c-a d) (a+b x)}+\frac {(d g-c h)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{h}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {\left (2 A B^2 (b c-a d) h^2 n\right ) \int x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b d}-\frac {\left (B^3 (b c-a d) h^2 n\right ) \int x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b d}-\frac {\left (2 A B^2 (b g-a h)^3 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b^2 h}-\frac {\left (B^3 (b g-a h)^3 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b^2 h}+\frac {\left (2 A B^2 (d g-c h)^3 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{d^2 h}+\frac {\left (B^3 (d g-c h)^3 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{d^2 h}-\frac {\left (2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b^2 d^2}-\frac {\left (B^3 (b c-a d) h (3 b d g-b c h-a d h) n\right ) \int \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b^2 d^2}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}+\frac {\left (A B^2 (b c-a d)^2 h^2 n^2\right ) \int \frac {x^2}{(a+b x) (c+d x)} \, dx}{b d}+\frac {\left (B^3 (b c-a d)^2 h^2 n^2\right ) \int \frac {x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{b d}-\frac {\left (2 A B^2 (b c-a d) (b g-a h)^3 n^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 h}-\frac {\left (2 B^3 (b c-a d) (b g-a h)^3 n^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{b^3 h}+\frac {\left (2 A B^2 (b c-a d) (d g-c h)^3 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{d^3 h}+\frac {\left (2 B^3 (b c-a d) (d g-c h)^3 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{d^3 h}+\frac {\left (2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^2}+\frac {\left (2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b^3 d^2}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}+\frac {2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {2 B^3 (b g-a h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (d g-c h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}+\frac {\left (A B^2 (b c-a d)^2 h^2 n^2\right ) \int \left (\frac {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{b d}+\frac {\left (B^3 (b c-a d)^2 h^2 n^2\right ) \int \left (\frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {a^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)}+\frac {c^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{b d}-\frac {\left (2 A B^2 (b c-a d) (b g-a h)^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^4 h}+\frac {\left (2 A B^2 (b c-a d) (d g-c h)^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{d^4 h}+\frac {\left (2 B^3 (b c-a d) (b g-a h)^3 n^3\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 h}+\frac {\left (2 B^3 (b c-a d) (d g-c h)^3 n^3\right ) \int \frac {\text {Li}_2\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{d^3 h}+\frac {\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 d^3}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac {A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac {A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac {2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {2 B^3 (b g-a h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (d g-c h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b g-a h)^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac {2 B^3 (d g-c h)^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {\left (B^3 c^2 (b c-a d) h^2 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b d^2}+\frac {\left (a^2 B^3 (b c-a d) h^2 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b^2 d}+\frac {\left (B^3 (b c-a d)^2 h^2 n^2\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b^2 d^2}+\frac {\left (2 A B^2 (b c-a d) (b g-a h)^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b^4 h}-\frac {\left (2 A B^2 (b c-a d) (d g-c h)^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{d^4 h}+\frac {\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{b^3 d^4}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac {A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac {A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac {2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {2 B^3 (b g-a h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (d g-c h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b g-a h)^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac {2 B^3 (d g-c h)^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}+\frac {\left (2 A B^2 (b c-a d) (b g-a h)^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{b^4 h}-\frac {\left (2 A B^2 (b c-a d) (d g-c h)^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{d^4 h}-\frac {\left (B^3 c^2 (b c-a d)^2 h^2 n^3\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b d^3}+\frac {\left (a^2 B^3 (b c-a d)^2 h^2 n^3\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 d}-\frac {\left (B^3 (b c-a d)^3 h^2 n^3\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^2}-\frac {\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{b^3 d^4}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac {A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac {A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac {2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac {B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {2 A B^2 (d g-c h)^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 h}-\frac {2 A B^2 (b g-a h)^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (b g-a h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (d g-c h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b g-a h)^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac {2 B^3 (d g-c h)^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {\left (B^3 c^2 (b c-a d)^2 h^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{b d^4}+\frac {\left (a^2 B^3 (b c-a d)^2 h^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^4 d}-\frac {\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{b^3 d^4}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac {A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac {A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac {2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac {B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {2 A B^2 (d g-c h)^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}-\frac {2 A B^2 (b g-a h)^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (b g-a h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (d g-c h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b g-a h)^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac {2 B^3 (d g-c h)^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}+\frac {\left (B^3 c^2 (b c-a d)^2 h^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{b d^4}-\frac {\left (a^2 B^3 (b c-a d)^2 h^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b^4 d}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac {A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac {A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac {2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac {B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {2 A B^2 (d g-c h)^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}-\frac {2 A B^2 (b g-a h)^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (b g-a h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (d g-c h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b g-a h)^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac {2 B^3 (d g-c h)^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}+\frac {\left (B^3 c^2 (b c-a d)^2 h^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{b d^4}-\frac {\left (a^2 B^3 (b c-a d)^2 h^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{b^4 d}\\ &=-\frac {A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac {A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac {A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac {A^3 (g+h x)^3}{3 h}-\frac {A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac {a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac {A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac {A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac {2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac {B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac {A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac {2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {2 A B^2 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac {2 A B^2 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac {B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac {B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac {A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac {B^3 (b g-a h)^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 h}-\frac {B^3 (d g-c h)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 h}+\frac {B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {2 A B^2 (d g-c h)^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 h}+\frac {B^3 c^2 (b c-a d) h^2 n^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}-\frac {2 A B^2 (b g-a h)^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 h}+\frac {a^2 B^3 (b c-a d) h^2 n^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 d}-\frac {2 B^3 (b g-a h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}-\frac {2 B^3 (d g-c h)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac {2 B^3 (b g-a h)^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac {2 B^3 (d g-c h)^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^3 h}\\ \end {align*}

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Mathematica [F]
time = 2.99, size = 0, normalized size = 0.00 \begin {gather*} \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3,x]

[Out]

Integrate[(g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3, x]

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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{2} \left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)

[Out]

int((h*x+g)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="maxima")

[Out]

A^2*B*h^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A^3*h^2*x^3 + 3*A^2*B*g*h*x^2*log((b*x + a)^n*e/(d*x + c)^n
) + A^3*g*h*x^2 + 3*(a*n*e*log(b*x + a)/b - c*n*e*log(d*x + c)/d)*A^2*B*g^2*e^(-1) - 3*(a^2*n*e*log(b*x + a)/b
^2 - c^2*n*e*log(d*x + c)/d^2 + (b*c*n - a*d*n)*x*e/(b*d))*A^2*B*g*h*e^(-1) + 1/2*(2*a^3*n*e*log(b*x + a)/b^3
- 2*c^3*n*e*log(d*x + c)/d^3 - ((b^2*c*d*n - a*b*d^2*n)*x^2*e - 2*(b^2*c^2*n - a^2*d^2*n)*x*e)/(b^2*d^2))*A^2*
B*h^2*e^(-1) + 3*A^2*B*g^2*x*log((b*x + a)^n*e/(d*x + c)^n) + A^3*g^2*x - 1/6*(2*(B^3*b^3*d^3*h^2*x^3 + 3*B^3*
b^3*d^3*g*h*x^2 + 3*B^3*b^3*d^3*g^2*x)*log((d*x + c)^n)^3 + 3*(2*(3*c*d^2*g^2*n - 3*c^2*d*g*h*n + c^3*h^2*n)*B
^3*b^3*log(d*x + c) - 2*(3*a*b^2*d^3*g^2*n - 3*a^2*b*d^3*g*h*n + a^3*d^3*h^2*n)*B^3*log(b*x + a) - 2*(A*B^2*b^
3*d^3*h^2 + B^3*b^3*d^3*h^2)*x^3 - (6*A*B^2*b^3*d^3*g*h + (a*b^2*d^3*h^2*n - (c*d^2*h^2*n - 6*d^3*g*h)*b^3)*B^
3)*x^2 - 2*(3*A*B^2*b^3*d^3*g^2 + (3*a*b^2*d^3*g*h*n - a^2*b*d^3*h^2*n - (3*c*d^2*g*h*n - c^2*d*h^2*n - 3*d^3*
g^2)*b^3)*B^3)*x - 2*(B^3*b^3*d^3*h^2*x^3 + 3*B^3*b^3*d^3*g*h*x^2 + 3*B^3*b^3*d^3*g^2*x)*log((b*x + a)^n))*log
((d*x + c)^n)^2)/(b^3*d^3) - integrate(-(3*A*B^2*b^3*c*d^2*g^2 + B^3*b^3*c*d^2*g^2 + (3*A*B^2*b^3*d^3*h^2 + B^
3*b^3*d^3*h^2)*x^3 + (B^3*b^3*d^3*h^2*x^3 + B^3*b^3*c*d^2*g^2 + (2*d^3*g*h + c*d^2*h^2)*B^3*b^3*x^2 + (d^3*g^2
+ 2*c*d^2*g*h)*B^3*b^3*x)*log((b*x + a)^n)^3 + (3*(2*d^3*g*h + c*d^2*h^2)*A*B^2*b^3 + (2*d^3*g*h + c*d^2*h^2)
*B^3*b^3)*x^2 + 3*(A*B^2*b^3*c*d^2*g^2 + B^3*b^3*c*d^2*g^2 + (A*B^2*b^3*d^3*h^2 + B^3*b^3*d^3*h^2)*x^3 + ((2*d
^3*g*h + c*d^2*h^2)*A*B^2*b^3 + (2*d^3*g*h + c*d^2*h^2)*B^3*b^3)*x^2 + ((d^3*g^2 + 2*c*d^2*g*h)*A*B^2*b^3 + (d
^3*g^2 + 2*c*d^2*g*h)*B^3*b^3)*x)*log((b*x + a)^n)^2 + (3*(d^3*g^2 + 2*c*d^2*g*h)*A*B^2*b^3 + (d^3*g^2 + 2*c*d
^2*g*h)*B^3*b^3)*x + 3*(2*A*B^2*b^3*c*d^2*g^2 + B^3*b^3*c*d^2*g^2 + (2*A*B^2*b^3*d^3*h^2 + B^3*b^3*d^3*h^2)*x^
3 + (2*(2*d^3*g*h + c*d^2*h^2)*A*B^2*b^3 + (2*d^3*g*h + c*d^2*h^2)*B^3*b^3)*x^2 + (2*(d^3*g^2 + 2*c*d^2*g*h)*A
*B^2*b^3 + (d^3*g^2 + 2*c*d^2*g*h)*B^3*b^3)*x)*log((b*x + a)^n) - (6*A*B^2*b^3*c*d^2*g^2 + 3*B^3*b^3*c*d^2*g^2
- 2*(3*c*d^2*g^2*n^2 - 3*c^2*d*g*h*n^2 + c^3*h^2*n^2)*B^3*b^3*log(d*x + c) + 2*(3*a*b^2*d^3*g^2*n^2 - 3*a^2*b
*d^3*g*h*n^2 + a^3*d^3*h^2*n^2)*B^3*log(b*x + a) + (B^3*b^3*d^3*h^2*(2*n + 3) + 2*A*B^2*b^3*d^3*h^2*(n + 3))*x
^3 + (6*(d^3*g*h*(n + 2) + c*d^2*h^2)*A*B^2*b^3 + (a*b^2*d^3*h^2*n^2 - ((n^2 - 3)*c*d^2*h^2 - 6*d^3*g*h*(n + 1
))*b^3)*B^3)*x^2 + 3*(B^3*b^3*d^3*h^2*x^3 + B^3*b^3*c*d^2*g^2 + (2*d^3*g*h + c*d^2*h^2)*B^3*b^3*x^2 + (d^3*g^2
+ 2*c*d^2*g*h)*B^3*b^3*x)*log((b*x + a)^n)^2 + (6*(d^3*g^2*(n + 1) + 2*c*d^2*g*h)*A*B^2*b^3 + (6*a*b^2*d^3*g*
h*n^2 - 2*a^2*b*d^3*h^2*n^2 + (2*c^2*d*h^2*n^2 - 6*(n^2 - 1)*c*d^2*g*h + 3*d^3*g^2*(2*n + 1))*b^3)*B^3)*x + 2*
(3*A*B^2*b^3*c*d^2*g^2 + 3*B^3*b^3*c*d^2*g^2 + (B^3*b^3*d^3*h^2*(n + 3) + 3*A*B^2*b^3*d^3*h^2)*x^3 + 3*((2*d^3
*g*h + c*d^2*h^2)*A*B^2*b^3 + (d^3*g*h*(n + 2) + c*d^2*h^2)*B^3*b^3)*x^2 + 3*((d^3*g^2 + 2*c*d^2*g*h)*A*B^2*b^
3 + (d^3*g^2*(n + 1) + 2*c*d^2*g*h)*B^3*b^3)*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b^3*d^3*x + b^3*c*d^2), x
)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="fricas")

[Out]

integral(A^3*h^2*x^2 + 2*A^3*g*h*x + A^3*g^2 + (B^3*h^2*x^2 + 2*B^3*g*h*x + B^3*g^2)*log((b*x + a)^n*e/(d*x +
c)^n)^3 + 3*(A*B^2*h^2*x^2 + 2*A*B^2*g*h*x + A*B^2*g^2)*log((b*x + a)^n*e/(d*x + c)^n)^2 + 3*(A^2*B*h^2*x^2 +
2*A^2*B*g*h*x + A^2*B*g^2)*log((b*x + a)^n*e/(d*x + c)^n), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)**2*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (g+h\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)^2*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3,x)

[Out]

int((g + h*x)^2*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3, x)

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